Question

The number of ways of arranging the letters $AAAAA, BBB, CCC, D, EE $ and $F$ in a row, if the letters $B$ are separated from one another, is equal to

• A. $\frac{13 !}{5 ! 3 ! 3 ! 2 !}$
• B. $\frac{14 !}{3 ! 3 ! 2 !}$
• C. $\frac{15 !}{\left(3 !\right)^{2} 2 ! 5 !}$
• D. ${}^{13}C_{3}^{}\times \frac{12 !}{5 ! 3 ! 2 !}$

Answer 1

Answer: (D)

All $AAAAA, \, CCC, \, D, \, EE, \, F$ can be arranged in $\frac{12 !}{5 ! 3 ! 2 !}$ ways.
Between the gaps $B$ can be arranged in ${}^{13}C_{3}^{}$ ways
Hence, the total number of ways $={}^{13}C_{3}\times \frac{12 !}{5 ! 3 ! 2 !}$